[proofplan] The proof is the classical weighted Hilbert-space argument. The Bochner-Kodaira identity gives an a priori estimate controlled by the Levi form of the strictly plurisubharmonic weight. That estimate makes a natural functional bounded on the range of $\bar\partial^*$. Riesz representation gives the desired solution, and exhaustion passes from smooth bounded subdomains to the full pseudoconvex domain. [/proofplan] [step:Prove the weighted basic estimate on smooth exhaustion domains] First assume $\Omega$ is smoothly bounded and work with compactly supported smooth $(0,1)$-forms $v$ in the domain of the weighted adjoint $\bar\partial^*_\varphi$. The Bochner-Kodaira identity gives \begin{align*} \|\bar\partial v\|_\varphi^2+\|\bar\partial^*_\varphi v\|_\varphi^2 \geq \int_\Omega \langle \mathcal{L}_\varphi v,v\rangle e^{-\varphi}\,d\mathcal{L}^{2n}. \end{align*} Pseudoconvexity makes the boundary term nonnegative, and strict plurisubharmonicity makes the Levi form $\mathcal{L}_\varphi$ positive definite. [/step] [step:Bound the functional determined by $f$] Let $f$ be $\bar\partial$-closed. For test forms $v$, Cauchy's inequality with respect to the Levi metric gives \begin{align*} |\langle f,v\rangle_\varphi|^2 \leq \left(\int_\Omega |f|^2_{\mathcal{L}_\varphi^{-1}}e^{-\varphi}\right) \left(\int_\Omega \langle\mathcal{L}_\varphi v,v\rangle e^{-\varphi}\right). \end{align*} The basic estimate bounds the second factor by $\|\bar\partial v\|_\varphi^2+\|\bar\partial^*_\varphi v\|_\varphi^2$. Since $\bar\partial f=0$, the functional vanishes on the $\bar\partial$-part and is bounded on the range of $\bar\partial^*_\varphi$. [/step] [step:Apply Riesz representation] Define a linear functional on $\operatorname{Ran}(\bar\partial^*_\varphi)$ by \begin{align*} \bar\partial^*_\varphi v\longmapsto \langle f,v\rangle_\varphi. \end{align*} The previous estimate shows it is bounded with norm at most \begin{align*} \left(\int_\Omega |f|^2_{\mathcal{L}_\varphi^{-1}}e^{-\varphi}\right)^{1/2}. \end{align*} By Hahn-Banach and Riesz, there is $u\in L^2(\Omega,e^{-\varphi})$ representing this functional. Thus \begin{align*} \langle u,\bar\partial^*_\varphi v\rangle_\varphi=\langle f,v\rangle_\varphi \end{align*} for all test $v$, which is precisely $\bar\partial u=f$ in the weak sense. The Riesz norm bound gives the stated estimate. [/step] [step:Pass to general pseudoconvex domains] For a general pseudoconvex domain, choose a smooth strictly pseudoconvex exhaustion $\Omega_j\Subset\Omega$. Apply the previous argument to $\Omega_j$ with regularized weights decreasing to $\varphi$ on compact subsets. The estimates are uniform on compact subsets, so weak compactness gives a subsequential limit $u\in L^2_{\mathrm{loc}}(\Omega)$. Closedness of $\bar\partial$ gives $\bar\partial u=f$, and lower semicontinuity gives the same weighted estimate on $\Omega$. [/step]